Michael McGovern also departed the Scottish Premier League club, signing for Ross County, after failing to make the breakthrough at Tannadice after his arrival from Celtic.
Zaluska, who was being presented to the media by Celtic on Monday, himself signed a pre-contract agreement in January to join the Glasgow club.
The Poland international, who had two-year spell at Tannadice, thanked Levein for taking a chance on him and predicted continuing good times at Tannadice.
"I would like to thank Dundee United and the coach and all the supporters, because it's thanks to them I am sitting here," who joined Celtic from Korona Kielce.
"They gave me the opportunity to become successful in Scotland.
"I think the Dundee United coach is that good. He loses good players every year, but he still manages to be successful."
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